Mathematical Sciences: Geometric Topology and the Fundamental Group

几何群是一个正常作用的群 在标准几何体上不连续且互补。 这 首席研究员试图了解 几何群的组合学： 组合数学的几何学吗 什么组合 几何群的性质 研究者 刚刚建立了一个组合特征， 余紧和有限体积双曲群在3维。 他还在创建计算机程序， 帮助学习递归计算结构 与双曲群和欧几里得群有关。 下一 研究者计划的步骤如下：（1）申请 尽可能多地用不同的方式来刻画他的性格 (Does这意味着在3维空间中负弯曲群 可以实现为共形群吗？ 瑟斯顿的 双曲化定理可以从刻画中推导出来吗？ 空间填充曲线可以这样研究吗？(2)使用电脑 程序来研究有趣的无限类的群体给出 通过生成器和关联器（例如，Coxeter群G（3，7，n）和 Conway's Fibonacci groups）。 本项目的工作使用了 几何学和群论之间微妙的相互作用 每个字段的状态。

A geometric group is a group acting properly discontinuously and cocompactly on a standard geometry. This principal investigator is attempting to understand the combinatorics of geometric groups: What are the implications of the geometry for the combinatorics? What combinatorial properties characterize a geometric group? The investigator has just established a combinatorial characterization of cocompact and finite-volume hyperbolic groups in dimension 3. He is also in the midst of the creation of computer programs to aid in studying the recursive computational structures associated with hyperbolic and Euclidean groups. The next steps in the investigator's plan are the following: (1) apply his characterization in as many different ways as possible (Does it imply that negatively curved groups in dimensions 3 can be realized as conformal groups? Can Thurston's hyperbolization theorem be deduced from the characterization? Can space-filling curves be so studied?); (2) use the computer programs to study interesting infinite classes of groups given by generators and relators (e.g., Coxeter's groups G(3,7,n) and Conway's Fibonacci groups). The work of this project uses the subtle interplay between geometry and group theory to advance the state of each field.